Mathematics, 16.06.2021 21:00 lauren21bunch

You decide to make a side business of selling face mask. The marginal cost for making a mask is $0.50 per mask. The total cost to to make 100 mask is hours is $62. You decide to sell the mask for $3 a mask. Required:
a. Find the linear cost function C(x).
b. Find the revenue function R(x).
c. How many mask must be made to break even. State in a complete sentence using the context of the problem.
d. Determine the interval of profit and loss. State in a complete sentence using the context of the problem

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Answer from: allieballey0727

Given:

Marginal cost for making 1 mask = $0.50

Total cost to make 100 masks = $62

Revenue per mask = $3

To find:

(a) Linear cost function C(x)

(b) Revenue function R(x)

(d) Interval of profit & loss

Solution:

(a) We know that linear cost function is given by,

Total cost = Fixed cost per unit + (Marginal cost per unit)*(Number of units)

Let 'x' denote the number of units

$\Rightarrow$ C(x) = b + mx

It is given that, m = $0.50

$\Rightarrow$ C(x) = b + 0.5x

It is also given that the total cost of making 100 masks is $62

$\Rightarrow$ C(100) = $62

$\Rightarrow$ b +0.5(100) = 62

$\Rightarrow$ b + 50 =62

$\Rightarrow$ b = 62 - 50

$\Rightarrow$ b = 12

$\Rightarrow$ C(x) = 12 + 0.5x

This is the linear cost function, C(x)

(b) We know that,

Total Revenue = Revenue per unit * Number of units

It is given that a mask is sold for $3, i.e., revenue per mask is $3

Let 'x' denote the number of units

Then the revenue function is given by,

R(x) = 3x

This is the revenue function, R(x)

(c) We know that, break even point refers to the point where total cost is equal to the total revenue. Thus, at break even point, we have,

C(x) = R(x)

$\Rightarrow$ 12 + 0.5x = 3x

$\Rightarrow$ 3x - 0.5x = 12

$\Rightarrow$ 2.5x = 12

$\Rightarrow x=\frac{12}{2.5}$

$\Rightarrow$ x = 4.8

Since, 'x' denotes the number of masks, it must be a whole number and not a fraction. Thus, we will round off our value to get the break even point as 5 masks.

That is, 5 masks must be made to break even.

(d) We know that the profit function is given as the difference of revenue function and cost function. That is, we have,

P(x) = R(x) - C(x)

$\Rightarrow$ P(x) = 3x - (12 + 0.5x)

$\Rightarrow$ P(x) = 3x - 12 - 0.5x

$\Rightarrow$ P(x) = 2.5x -12

Now, we know that there is profit when the value of the profit function is positive & there is loss when the value of profit function is negative. Thus, we can calculate the intervals of profit and loss by finding the intervals where the profit function is positive & negative respectively. Alternatively, since the break even point denotes the point where the value of the profit function is 0, we can find the intervals of profit and loss as the intervals greater than and lesser than the break even point respectively.

That is, since the break even point is x = 4.8, the interval of profit is given as x > 4.8 & the interval of loss is given as x < 4.8

Taking into account that 'x' denotes the number of masks and thus must be a whole number, we have the intervals of profit and loss as,

Loss:= $x \in [0,4]$